平方和和多项式微积分中的单项大小与位复杂度

Tuomas Hakoniemi
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引用次数: 3

摘要

本文研究了在有理数$({\text{PCR}}/\mathbb{Q})$上多项式微积分解析中平方和(SOS)的一元大小与位复杂度之间的关系。我们证明了在布尔变量上存在一组多项式约束Qn,它同时具有SOS和${\text{PCR}}/\mathbb{Q}$ 2次的反驳,因此只有多项式多个单项式,但是对于任何SOS或${\text{PCR}}/\mathbb{Q}$反驳必须具有指数位复杂度,当有理系数用它们的简化分数表示为二进制时。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Monomial size vs. Bit-complexity in Sums-of-Squares and Polynomial Calculus
In this paper we consider the relationship between monomial-size and bit-complexity in Sums-of-Squares (SOS) in Polynomial Calculus Resolution over rationals $({\text{PCR}}/\mathbb{Q})$. We show that there is a set of polynomial constraints Qn over Boolean variables that has both SOS and ${\text{PCR}}/\mathbb{Q}$ refutations of degree 2 and thus with only polynomially many monomials, but for which any SOS or ${\text{PCR}}/\mathbb{Q}$ refutation must have exponential bit-complexity, when the rational coefficients are represented with their reduced fractions written in binary.
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